Partial Fraction Decomposition by Division Sidney H

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Partial Fraction Decomposition by Division Sidney H. Kung ([email protected]) University of North Florida, Jacksonville FL 32224

In this note, we present a method for the partial fraction decomposition of two algebraic functions: (i) f (x)/(ax + b)t and (ii) f (x)/(px2 + qx + r)t ,where f (x) is a polynomial of degree n, t is a positive integer, and px2 + qx + r is an irreducible (q2 < 4pr) quadratic polynomial. Our algorithm is relatively simple in comparison with those given elsewhere [1, 2, 3, 4, 5, 6, 7, 8]. The essence of the method is to use repeated division to re-express the numerator polynomial in powers of the normalized denominator. Then upon further divisions, we obtain a sum of partial fractions in the i 2 j form Ai /(ax + b) or (B j x + C j )/(px + qx + r) for the original function. For (i), we let c = b/a, and express f (x) as follows:

n n−1 2 f (x) = An(x + c) + An−1(x + c) +···+ A2(x + c) + A1(x + c) + A0

n−1 n−2 = (x + c)[An(x + c) + An−1(x + c) +···+ A2(x + c) + A1]+A0, where each Ai is a real coefﬁcient to be determined. Then the remainder after we divide f (x) by x + c gives the value of A0. The quotient is

n−2 n−3 q1(x) = (x + c)[An(x + c) + An−1(x + c) +···+ A3(x + c) + A2]+A1.

If we now divide q1(x) by x + c, we see that the next remainder is A1 and the quotient is

n−3 n−4 q2(x) = (x + c)[An(x + c) + An−1(x + c) +···+ A3]+A2.

Continuing to divide in this manner n − 1 times, we get the quotient qn−1(x) = An(x + c) + An−1. Finally, dividing qn−1(x) by x + c, we obtain the last two coefﬁ- cients, An−1 and An. Thus, it follows that f (x) 1 A A − A A = n + n 1 +···+ 1 + 0 . (1) (ax + b)t at (x + c)t−n (x + c)t−n+1 (x + c)t−1 (x + c)t

For example, to ﬁnd the partial fraction decomposition of (x 4 + 2x 3 − x 2 + 5)/ 5 (2x − 1) ,weusec =−1/2 and perform synthetic division to obtain A0 through An.

1/2) 1 2 −10 5 1/25/41/81/16

15/21/41/8 | 81/16 ⇐ A0 1/23/27/8

13 7/4 | 1 ⇐ A1 1/27/4

17/2 | 7/2 ⇐ A2 1/2

A4 ⇒ 14⇐ A3

132 c THE MATHEMATICAL ASSOCIATION OF AMERICA Substituting the coefﬁcients into (1), we have 4 + 3 − 2 + / x 2x x 5 = 1 1 + 4 + 7 2 + 1 (2x − 1)5 25 (x − 1/2) (x − 1/2)2 (x − 1/2)3 (x − 1/2)4 / + 81 16 (x − 1/2)5 / / / = 1 16 + 1 2 + 7 8 2x − 1 (2x − 1)2 (2x − 1)3 / / + 1 2 + 81 16 . (2x − 1)4 (2x − 1)5

For (ii), we let u = q/p, v = r/p, and express f (x) in the following form:

2 (n−1)/2 2 (n−3)/2 f (x) = B(n−1)/2(x + ux + v) + B(n−3)/2(x + ux + v) +···

2 + B1(x + ux + v) + B0, where each coefﬁcient Bk , k = 0, 1,... ,(n − 1)/2, is a linear function of x,and whereweassumethatn ≤ 2t − 1. In this case, dividing f (x) and each successive quotient by x 2 + ux + v as described above, we obtain f (x) 1 B( − )/ = n 1 2 (2) (px2 + qx + r)t pt (x 2 + ux + v)t−(n−1)/2 + B(n−3)/2 +···+ B0 . (x 2 + ux + v)t−(n−3)/2 (x 2 + ux + v)t

For instance, take the rational function (x 5 − 4x 4 + 3x 2 − 2)/(x 2 − x + 2)3.Then u =−1andv = 2. Since (n − 1)/2 = 2, we let x 5 − 4x 4 + 3x 2 − 2 = (Mx + N)(x 2 − x + 2)2 + (Kx + L)(x 2 − x + 2) + Ix + J.

Since most students are not familiar with the synthetic division technique when the divisor is a quadratic polynomial, long division can be used in place of the following computation to ﬁnd the coefﬁcients I , J, K , L, M,andN.

1 −403 0 −2 | 1 −3 −54 1 −12 | 1 −12 −3 −23 −336 −59 0 −55−10 410 −2 4 −48 14 −10 IJ

VOL. 37, NO. 2, MARCH 2006 THE COLLEGE MATHEMATICS JOURNAL 133 MN 1 −3 −54| 1 −2 1 −12 | 1 −12 −2 −74 −22−4 −98 KL

Substituting the coefﬁcients in (2) (note that t − (n − 1)/2 = 1) gives

5 − 4 + 2 − − − + − x 4x 3x 2 = x 2 + 9x 8 + 14x 10 . (x 2 − x + 2)3 x 2 − x + 2 (x 2 − x + 2)2 (x 2 − x + 2)3

Note also that x 2 − x + 2 = (x − 1/2)2 + 4/7. On the right hand side of the above expression, replacing the coefﬁcients −2, 8, and −10 in the numerators by −2 + 1/2M, 8 + 1/2M, and −10 + 1/2M, respectively, we get

5 − 4 + − ( − / ) − / − ( − / ) + / x 4x 3x 2 = x 1 2 3 2 + 9 x 1 2 7 4 ((x − 1/2)2 + 7/4)3 (x − 1/2)2 + 7/4 ((x − 1/2)2 + 7/4)2 ( − / )2 − + 14 x 1 2 3 . ((x − 1/2)2 + 7/4)3

This last expression is an easily antidifferentiable form.

References

1. S. Burgstahler, An alternative for certain partial fractions, College Math. J. 15 (1984) 57–58. 2. X.-C. Huang, A short cut to partial fractions, College Math. J. 22 (1991) 413–415. 3. P. T. Joshi, Efﬁcient techniques for partial fractions, College Math. J. 14 (1983) 110–118. 4. J. E. Nymann, An alternative for partial fractions (part of the time), College Math. J. 14 (1983) 60–61. 5. P. Schultz, An algebraic approach to partial fractions, College Math. J. 14 (1983) 346–348. 6. M. R. Spiegel, Partial fractions with repeated linear or quadratic factors, Amer. Math. Monthly 57 (1950) 180–181. 7. T. N. Subramaniam and D. E. G. Malm, How to integrate rational functions, Amer. Math. Monthly 99 (1992) 762–772. 8. J. Wiener, An algebraic approach to partial fractions, College Math. J. 17 (1986) 71–72.

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An Elegant Mode for Determining the Mode D.S. Broca ([email protected]), Indian Institute of Management Kozhikode, Kozhik- ode 673 571, Kerala, India

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134 c THE MATHEMATICAL ASSOCIATION OF AMERICA